Checkpoint: Spin Dynamics

This checkpoint ties together everything in the module: the spin–qubit mapping, the rotation operator, Rabi driving, and Larmor precession. You will assemble a short pulse sequence of the kind used in nuclear magnetic resonance and qubit control.

The physics of the sequence

Start with a spin in $|0\rangle$ — fully polarized along $+z$ . Two operations follow.

1. A $\pi/2$ pulse about $x$ . On resonance, a brief transverse drive rotates the spin about the $x$ -axis. Choosing the pulse area $\Omega t = \pi/2$ applies

R_X\!\left(\frac{\pi}{2}\right) = \cos\frac{\pi}{4}\,I - i\sin\frac{\pi}{4}\,\sigma_x = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix},

which tips the polarization from $+z$ down into the equatorial plane, creating a coherent superposition with $\langle S_z\rangle = 0$ .

2. Free Larmor precession by $\pi/2$ . Left in the static field, the equatorial spin precesses about $z$ . Over a precession angle $\omega_L t = \pi/2$ the evolution is

R_Z\!\left(\frac{\pi}{2}\right) = \operatorname{diag}\!\left(e^{-i\pi/4},\, e^{+i\pi/4}\right),

rotating the Bloch arrow within the equatorial plane.

Order matters

These two rotations are about different axes, and rotations about different axes do not commute — the very fact encoded by $[\sigma_x, \sigma_z] = -2i\sigma_y$ . Applying $R_Z$ before $R_X$ would produce a physically different state. The sequence below is $R_X$ first, then $R_Z$ .

Try it

Build the sequence on one qubit: apply $R_X(\pi/2)$ first, then $R_Z(\pi/2)$ . The grader checks the exact resulting state vector, so both the gates and their order must be correct.

Run your code to see the quantum state.